There's going to be some hobbyists doing math in their basements, but there's unlikely to be a future for professional mathematicians.
That's exactly what the Simons foundation was set up to reject, and it's one of most important funding streams for math
The proof with the 11/12 constant, the simplest one, has already prompted what looks like chagrin from some. It's an unexpected combination of known results/techniques, a combination that apparently fell into a blind spot so it wasn't picked up before (even though it could have been, even 20 years ago).
I think these results are showing there was a lot of low hanging fruit that human mathematicians didn't have the breadth or mental bandwidth to find.
To address what you actually meant: it's possible but specifications are way easier to verify.
In general, once the proof has been formalized, it's supposed to be the proof of the correct theorem (that's what the Lean proof-checker checks)
the obverse would be to assert that Lean (or any other theorem prover) is infallible under all circumstances, present and future - an even weightier claim in my opinion.
But on the other hand, agree with you that the chances that one its bugs causes a solved proofs to be wrong is probably very, very low. Yeah, each version of Lean obviously must be run against hundreds if not thousands tests of existing proofs to ensure it's working correctly...
... Although, one thing that is a little worrying is that some mathematicians have said the Lean code generated in these OpenAI proofs is supposedly really complex and undecipherable. It might be pushing Lean to corner cases that are not well tested.
So as long as the "somewhat simple" core that checks to proof is correct, it doesn't matter if the "programming language" has bugs: either in the end the "proof assembly code" generated by the program is right and is validated by the core, or it's not, and you need to revise your proof, or to check for either a bug in the programming language (it didn't generate the proof you were expecting), or in very rare case detect a bug in the simple "core"
I never advanced in math past high school, but my assumption would be that we as a society aren't paying mathematicians to have fun[0]. The discoveries mathematicians make must translate into real improvements in engineering and other life improving fields. So why would we pay mathematicians to work on math much the same way, if a new machine-led way makes these discoveries faster and cheaper?
[0] Being a little facetious. I know I don't know what I don't know.
With math you never know up front what 'real improvements in engineering and other life improving fields' will require some key bit of it that someone interested in pure math came up with a decade or two (or 10) before we found an application.
Obviously. In my ideal world even the janitors at the same institutions would have a blast, it's just not usually the end-goal reason humans are hired.
>With math you never know up front what 'real improvements in engineering and other life improving fields' will require some key bit of it that someone interested in pure math came up with a decade or two (or 10) before we found an application.
Right, but would it have made a difference for the application layer if that key bit of "pure math" had been discovered by a human (that had a great time) or a machine? That's my point about games being a bad analogy.
For instance, almost all of cryptography is applied math. If your opponent understands the math better than you do you might end up with your secrets compromised.
Math is about as 'pure' a science as there is, the touchstone of applicability is missing the point entirely, it is first and foremost about understanding things at one of the most fundamental levels (even more fundamental than physics) that there is.
Nobody cares all that much about the discovery (except that we name things for the person that discovers it, it's sort of analogous to the discovery of a new piece of land). And given that the pay usually sucks this is a nice thing to be able to do. What mathematicians usually do really care about is their ability to understand the discovery and that this discovery has the proper foundations (other pieces of math that are also understood). It doesn't really matter what the answer is if you don't actually fully understand the question or the foundation the answer builds on.